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\def\Z{\mathbb{Z}}
\def\R{\mathbb{R}}
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\def\N{\mathbb{N}}

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\def\P{\mathcal{P}}

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\noindent
{\tiny \url{http://drorbn.net/18-327/ThingsYouShouldKnow.html}}
\newline{\footnotesize
  \href{http://www.math.toronto.edu/~drorbn/Copyleft/}{\textcopyleft} $\mid$
  \href{http://www.math.toronto.edu/~drorbn/}{Dror Bar-Natan}:
  \href{http://www.math.toronto.edu/~drorbn/classes/index.html}{Classes}:
  \href{http://www.math.toronto.edu/~drorbn/classes/index.html#1819}{2018-19}:
  \href{http://www.math.toronto.edu/~drorbn/classes/18-327-Topology/}{MAT327}:
}
\hfill{\LARGE\bf Things You Should Already Know}


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\begin{multicols}{2} \raggedcolumns

I will assume that you are familiar with all of the terms and symbols on this handout. Our first tutorials will go over everything here, just in case something is missing.

\section{Basic Set Theory}

In the following, $A,B,X,Y$ are sets, $I$ is an indexing set and $\set{A_\a}{\a \in I}$ and $\set{B_\a}{\a \in I}$ are families of sets indexed by $I$.

\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
	\item
		Empty set:
			$\emptyset$,
		the set with no elements.
	\item
		Subset:
			$A \sse B$ means ``$x \in A \implies x \in B$''
	\item
		Union:
			$A \cup B \defas \set{x}{x \in A \text{ or } x \in B}$
	\item
		Intersection:
			$A \cap B \defas \set{x}{x \in A \text{ and } x \in B}$
	\item
		Complement:
			If $A \sse X$, then $X \setminus A \defas \set{x}{x \in X \text{ and } x \notin A}$
	\item
		Indexed union:
			$\bigcup_{\a \in I} A_\a \defas \set{x}{\exists \a \in I, \; x \in A_\a}$
	\item
		Indexed intersection:
			$\bigcap_{\a \in I} A_\a \defas \set{x}{\forall \a \in I, \, x \in A_\a}$
	\item
		Cartesian product of two sets:
			$X \times Y \defas \set{(x,y)}{x \in X, \; y \in Y}$
    \item Powers of sets: $Y^X$ is the set of all function $f\colon X\to Y$.
	\item
		The power set of $X$:
			$\P(X) \defas \set{A}{A \sse X}\leftrightarrow \{0,1\}^X$.
\end{itemize}









\section{Functions}

In the following, let $X$ and $Y$ be sets, and let $f: X \to Y$ be a function.

\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
	\item $X$ is the \ul{domain} of $f$.
	\item $Y$ is the \ul{target space} or \ul{codomain} of $f$.
	\item $f(X) \!=\! \set{f(x)}{x \in X} \!\sse\! Y$ is the \ul{range} or \ul{image} of $f$.
	\item $f$ is \ul{injective} (or \ul{one-to-one}, or an \ul{injection})
		\[
			\forall a, b \in X, \quad f(a) = f(b) \implies a = b.
		\]
	\item $f$ is \ul{surjective} (or \ul{onto}, or a \ul{surjection}) if its range is its entire codomain.
	\item $f$ is \ul{bijective} (or a \ul{bijection}) if it is both injective and a surjective.
	\item The composition of two injective functions is again injective.
	\item The composition of two surjective functions is again surjective.
	\item The composition of two bijective functions is again bijective.
	\item Given a subset $B \sse Y$, the \ul{preimage} of $B$ is the set $f^{-1}(B) \defas \set{x \in X}{f(x) \in B}$.
	\item If $f$ is an injection with range $Y$, then its inverse function $f^{-1}: Y \to X$ is (1) a function; and (2) bijective.
\end{itemize}










\section{De Morgan's Laws and some Further Relations}

The following two expressions are generalized versions of what are called De Morgan's Laws. They describe how unions and intersections interact with complementation.
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
	\item $\ds{X \setminus \left( \bigcup_{\a \in I} A_\a \right) = \bigcap_{\a \in I} (X \setminus A_\a )}$
	\item $\ds{X \setminus \left( \bigcap_{\a \in I} A_\a \right) = \bigcup_{\a \in I} (X \setminus A_\a)}$
\end{itemize}

\

\no The following are elementary facts about how functions interact with operations on subsets of their domains, codomains and ranges. Throughout the following, let $X$ and $Y$ be sets, let $f: X \to Y$ be a function, and let $A,B \sse X$ and $C,D \sse Y$.

\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
	\item $A \sse B$ implies $f(A) \sse f(B)$
	\item $C \sse D$ implies $f^{-1}(C) \sse f^{-1}(D)$
	\item $f(A \cup B) = f(A) \cup f(B)$
	\item $f^{-1}(C \cup D) = f^{-1}(C) \cup f^{-1}(D)$
	\item $f(A \cap B) \sse f(A) \cap f(B)$
	\item $f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)$
	\item $f(A) \setminus f(B) \sse f(A \setminus B)$
	\item $f^{-1}(C \setminus D) = f^{-1}(C) \setminus f^{-1}(D)$
	\item $f(X \setminus f^{-1}(Y \setminus C)) \sse C$
	\item $A \sse f^{-1}(f(A))$, (with equality if $f$ is injective)
	\item $f(f^{-1}(C)) \sse C$, (with equality if $f$ is surjective)
	\item $f^{-1}(Y \setminus C) = X \setminus f^{-1}(C)$
\end{itemize}










\section{Countability and Uncountability}

We will spend some time on this in class, but I do expect these words to be familiar to you.

\begin{dfn}
	A set $A$ is said to be \ul{countably infinite} if there exists a bijection $f: \N \to A$. A set $A$ is said to be \ul{countable} if it is finite or countably infinite. If $A$ is infinite but not countably infinite, $A$ is said to be \ul{uncountable}.
\end{dfn}

The following theorem gives some equivalent conditions for being countable:

\begin{thm} For an infinite set $A$, the following are equivalent:
	\begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
		\item $A$ is countable.
		\item There is an injection $f: A \to \N$.
		\item There is a surjection $g: \N \to A$.
	\end{enumerate}
\end{thm}

\no\textbf{Fact}: The following sets are countable:

\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
	\item $\N, \Z, \Q$, the set of algebraic numbers.
	\item Any infinite subset of a countable set.
	\item The Cartesian product of two countable sets (and, inductively, the Cartesian product of a finite number of countable sets).
	\item The union of finitely many countable sets.
	\item The union of a countable collection of countable sets.
	\item The countable union of some countable sets and some finite sets.
\end{itemize}

\no\textbf{Fact}: The following sets are uncountable:

\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
	\item $\R$, $\R \setminus \Q$ (the irrational numbers), the set of non-algebraic numbers (i.e. the set of transcendental numbers), $\R^n$.
	\item Any superset of an uncountable set.
	\item The power set of any infinite set (countable or otherwise), e.g. $\P(\N)$.
	\item The set $\N^\N$ of functions from $\N$ to $\N$.
\end{itemize}


\section{\texorpdfstring{Selected Basic Facts About $\R$}{Selected Basic Facts About R}}

First recall: $\N \subset \Z \subset \Q \subset \R$. (For us, $0 \notin \N$.)

\vspace*{\baselineskip}

\no\textbf{Fact}: Between any two distinct real numbers:

\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
	\item There are infinitely many rational numbers.
	\item There are infinitely many irrational numbers.
\end{itemize}

\no\textbf{Fact}: Here are some useful facts from calculus:

\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
	\item $\ds{\bigcup_{n \in \N} [\tfrac{1}{n}, 1] = (0,1]}$.
	\item $\ds{\bigcup_{n \in \N} [0, n] = [0, \infty)}$.
	\item $\ds{\sum_{n \in \N} 2^{-n} = 1}$.
\end{itemize}

\section{Acknowledgement}

This document was modified from a document by \href{http://boolesrings.org/mpawliuk/}{Micheal Pawliuk} and by \href{http://www.math.toronto.edu/~ivan/}{Ivan Khatchatourian} and used with their permission.
Thanks, Micheal and Ivan!

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